This is a continuation of last week's talk of quasicrystals in possibly non-commutative lcsc groups. We introduce the notion of Gelfand pairs which are a powerful tool to develop a spherical Fourier theory on the spaces we have in mind. The corresponding abstract Fourier transform allows us to define the spherical diffraction measure for non-commutative quasicrystals. Generalizing the classical (abelian) case, we explain why regular cut-and-project sets arising from cocompact lattices amount to pure point diffraction. Joint work with Michael Björklund and Tobias Hartnick.